How to animate A curve segment?
You can 'Hook' the control point of the curve by pressing Ctrl+H while in edit mode, with the point of the curve you want to move selected and choosing 'Hook to new object'. Adding the hook via this shortcut is just a quick way of setting up the hook modifier, which will be present on the curve after the shortcut is pressed.

This will create an 'Empty' object which can be animated. The control point of the curve will be parented to the position of the empty and will follow the empty wherever you animate it to.